| L(s) = 1 | + 1.02·2-s − 0.959·4-s − 2.96·5-s + 0.816·7-s − 3.01·8-s − 3.02·10-s − 4.92·11-s + 1.06·13-s + 0.833·14-s − 1.16·16-s + 0.905·17-s − 0.776·19-s + 2.83·20-s − 5.02·22-s − 6.93·23-s + 3.76·25-s + 1.08·26-s − 0.783·28-s + 1.72·29-s − 5.90·31-s + 4.85·32-s + 0.923·34-s − 2.41·35-s + 5.58·37-s − 0.791·38-s + 8.93·40-s − 5.11·41-s + ⋯ |
| L(s) = 1 | + 0.721·2-s − 0.479·4-s − 1.32·5-s + 0.308·7-s − 1.06·8-s − 0.955·10-s − 1.48·11-s + 0.295·13-s + 0.222·14-s − 0.290·16-s + 0.219·17-s − 0.178·19-s + 0.634·20-s − 1.07·22-s − 1.44·23-s + 0.752·25-s + 0.213·26-s − 0.148·28-s + 0.319·29-s − 1.06·31-s + 0.857·32-s + 0.158·34-s − 0.408·35-s + 0.918·37-s − 0.128·38-s + 1.41·40-s − 0.798·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6755649607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6755649607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 - 1.02T + 2T^{2} \) |
| 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 - 0.816T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 - 0.905T + 17T^{2} \) |
| 19 | \( 1 + 0.776T + 19T^{2} \) |
| 23 | \( 1 + 6.93T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 + 0.255T + 43T^{2} \) |
| 47 | \( 1 + 9.75T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 9.97T + 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 - 7.97T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 - 7.82T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.959642853456224249208540440539, −7.70541931444764219416372318153, −6.49899240097784459777528434302, −5.81418516352230924106895064988, −4.87454533914508804336642755759, −4.62411585556633856916294075178, −3.58461556819143646221133861819, −3.24827186315376035765485550216, −2.02785769813661891726326953084, −0.36920730723501297867849695509,
0.36920730723501297867849695509, 2.02785769813661891726326953084, 3.24827186315376035765485550216, 3.58461556819143646221133861819, 4.62411585556633856916294075178, 4.87454533914508804336642755759, 5.81418516352230924106895064988, 6.49899240097784459777528434302, 7.70541931444764219416372318153, 7.959642853456224249208540440539
